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Pascal's and Huygens's Game-Theoretic Foundations for Probability sarton2018_vol.32.book Page 117 Monday, September 30, 2019 10:37 AM Pascal’s and Huygens’s game-theoretic foundations for probability Glenn Shafer Blaise Pascal and Christiaan Huygens developed game-theoretic founda- tions for the calculus of chances – foundations that replaced appeals to frequency with arguments based on a game’s temporal structure. Pascal argued for equal division when chances are equal. Huygens extended the argument by considering strategies for a player who can make any bet with any opponent so long as its terms are equal. These game-theoretic foundations were disregarded by Pascal’s and Huygens’s 18th century successors, who found the already established foundation of equally frequent chances more conceptually relevant and mathematically fruitful. But the game-theoretic foundations can be devel- oped in ways that merit attention in the 21st century. 1. The calculus of chances before Pascal and Fermat We are often told that probability theory began with an exchange of letters in 1654 between Blaise Pascal (1623-1662) and Pierre Fermat (1607- 1665). As Florence Nightingale David put it, The name of Blaise Pascal is always linked with that of Fermat as one of the “joint discoverers” of the probability calculus.1 1 [1], p. 75. 117 sarton2018_vol.32.book Page 118 Monday, September 30, 2019 10:37 AM 118 We can trace this attribution back to Laplace, who told his students at the École Normale in 1795 that probability theory owes its birth to two French geometers of the 17th century.2 Laplace repeated these words in 1812, in the first edition of his Théorie analytique,3 but he tempered them two years later in the history of proba- bility theory with which he concluded his Essai philosophique: For quite a long time, people have ascertained the ratios of favourable to unfavourable chances in the simplest games; stakes and bets were fixed by these ratios. But before Pascal and Fermat, no one gave principles and methods for reducing the matter to calculation, and no one had solved prob- lems of this type that were even a little complicated. So we should attribute to these two great geometers the first elements of the science of probabili- ties …4 Many of Laplace’s successors have found the nuances unnecessary. Lacroix, for example, began his 1816 probability textbook with this unqualified attribution: The probability calculus, invented by Pascal and Fermat, has never since ceased exciting the interest and exercising the wisdom of their most illus- trious successors …5 Similar unqualified statements by mathematicians and historians of math- ematics abound, throughout the 19th and 20th centuries and up to the present day. But Laplace was surely correct when he conceded that people had been counting chances and using the counts to fix stakes and bets long before Pascal and Fermat. Counting chances People have been making finely balanced dice for millennia, and they have probably been counting the chances for throws of these dice for just as 2 Apparently not published at the time, Laplace’s lecture was reproduced on pp. 146-177 of Volume XIV of his complete works [2]. The words translated here come at the end of the lecture. Except when otherwise noted, all translations are mine. 3 [3], p. 3. 4 [4], p. 89. 5 [5], p. iii. sarton2018_vol.32.book Page 119 Monday, September 30, 2019 10:37 AM 119 long. But the earliest documentary evidence for such counting appears to be the Latin poem De Vetula, probably written around 1250 by a teacher of the quadrivium (arithmetic, geometry, astronomy, and music) at the University of Paris.6 A long poem, touching on philosophical, religious, and scientific topics, De Vetula begins by warning its readers against the temptations of erotic love and gambling. In the case of gambling, the author warns that a gambler faces ruin even if he knows how to count chances, then proceeds to count them anyway for the sum of the points on three dice. There are 216 chances, he explains, all of equal force and frequency. But the sum of the points can range from 3 to 18, and these 16 possibilities have unequal force and frequency. There are 108 chances that the sum will be between 3 and 10, distributed unequally: Sum of points 3 4 5 6 7 8 9 10 Total # of chances 1 3 6 10 15 21 25 27 108 There are another 108 chances that the sum will be between 11 and 18, distributed similarly: Sum of points 18 17 16 15 14 13 12 11 Total # of chances 1 3 6 10 15 21 25 27 108 David Bellhouse has called De Vetula a “medieval bestseller”. It was often quoted. Nearly 60 manuscript copies survive. The first printed edition appeared in about 1475. Not everyone who reproduced it understood how the author counted chances. But some did, including the editors of editions printed in 1479, 1534, and 1662. There are at least two other surviving documents in which mathematicians counted the chances for dice before 1654: a book by Cardano, who died in 1576, and a letter by Galileo, who died in 1642. Neither appeared in print in its author’s lifetime. Cardano’s Liber de Ludo Aleae was published in his collected works in 1663,7 and Galileo’s letter appeared in his collected works in 1718.8 Both Cardano and Galileo counted the chances for the sum 6 [6], [7], [8], [9]. 7 Geralamo Cardano, Liber de Ludo Aleae, in [10], volume 1, pp. 262-276. English translation in [11], pp. 182-241. 8 [12], pp. 591-594. English translation [1], pp. 192-195. sarton2018_vol.32.book Page 120 Monday, September 30, 2019 10:37 AM 120 of points on three dice. As Bellhouse has pointed out, Cardano’s presenta- tion suggests that he may have been influenced directly by De Vetula, whereas Galileo obtains the counts in a different way.9 As mathematics developed, mathematicians’ ability to count chances improved. Galileo mentions that the number of equally frequent chances is multiplied by 6 every time a die is added to the throw. There are 216 equal chances in the case of three dice because 6 × 6 × 6 = 216. The author of De Vetula had not mentioned this. Fixing stakes and bets The whole point of counting chances is to use them to fix stakes and bets. The author of De Vetula does not bother to explain how this is done, but readers adept in mathematics would have known what to do: use the rule of three. The universities of medieval Europe prepared young men for careers in the priesthood, law and medicine. To learn practical mathematics, you went elsewhere – to teachers who prepared young men to work in trade. We know what these teachers taught, because countless of their manuals – commercial arithmetics, we call them – have survived. This being a lecture in honour of George Sarton, I pause to recall Sarton’s interest in these manuals. As he pointed out in 1933, they were being written in both Arabic and in Spanish in Spain in the 11th century.10 They spread throughout Europe as trade developed.11 The rule of three was the main tool of the commercial arithmetics. After learning how to add, subtract, divide and multiply, merchants and their clerks need to understand proportions. If you buy 15 bushels of wheat for 10 shillings, what price should you charge someone else for 3 bushels? For us, this is a matter of algebra: 15/3 = 10/x, and so x = 2 shillings. But al- Khwarizmi’s 9th-century algebra was all in words, and the medieval commercial arithmetics still had only words. Algebra with symbols emerged only in the Renaissance. It was largely developed by the authors of commercial arithmetics – the Italian abacus masters and the German 9 [13]. 10 [14], [15]. 11 See for example [16]. sarton2018_vol.32.book Page 121 Monday, September 30, 2019 10:37 AM 121 reckoning masters. But even in the 19th century, commercial arithmetics emphasized the non-symbolic rule of three, deploying it in problem after problem in which you find an unknown fourth number in a proportion from three that are known, problems about trading in goods, dividing profits, changing currencies, pricing alloys, etc., etc. Occasionally, for fun, an author might throw in a problem about a game. Here are two questions that could have been answered by anyone who was adept at the rule of three and could count the chances for three dice. Q1. Three dice are to be thrown repeatedly until either a 9 or a 15 appears. Player A bets on 9 and Player B bets on 15. Player A puts 5 shillings on the table. How much should Player B put on the table? Q2. What should Player A pay in order to win 80 shillings if he throws an 11 on a single throw of three dice? Permitting ourselves a bit of algebra rather than trying to imitate a 13th- century abacus teacher’s use of the rule of three, we can answer these ques- tions as follows. A1. To answer Q1, we recall that there are 25 chances of throwing a 9 and only 10 chances of throwing a 15. The chances have equal frequency. So Player B wins 10 times for every 25 times Player A wins. Player A has put 5 shillings on the table for Player B to win. If we write x for the amount Player B puts on the table, then Player B wins 10 × 5 shil- lings every time Player A wins 25 × x.
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